Simplifying Linear Algebra: Techniques, Solutions, and Examples

I have a rectangle with center $(x_1,y_1)$ and sides $a,b$ where side $a$ is parallel to axis $Ox$. I want to find a transformation matrix that:
a) converts this rectangle into a square with the same center and side $d$.
b) reflects the rectangle with mirror axis the line $y=sx+c$.

A linear transformation $T:<!--\; :\; -->{\mathbb{R}}^3\to <!--\; \backslash rightarrow\; -->{\mathbb{R}}^2$ whose matrix is
$\left(\begin{array}{ccc}1& 3& 3\\ 2& 6& -<!--\; -\; -->3.5+k\end{array}\right)$
is onto if and only if $k\ne <!--\; \ne \; -->$

Let $V$ be inner product space.
Let $e_1,...,e_n$ an orthonormal basis for $V$
Let $z_1,...,z_n$ an orthonormal basis for $V$
I have to show that the matrix represents the transformation matrix between $e_1,...,e_n$ to $z_1,...,z_n$ is unitary.

If
$A=\left[\begin{array}{ccc}1& -<!--\; -\; -->1& 2\\ -<!--\; -\; -->2& 1& -<!--\; -\; -->1\\ 1& 2& 3\end{array}\right]$
is the matrix representation of a linear transformation
$T:P_2(x)\to <!--\; \backslash rightarrow\; -->P_2(x)$
with respect to the bases $\{1-<!--\; -\; -->x,x(1-<!--\; -\; -->x),x(1+x)\}$ and $\{1,1+x,1+x^2\}$ then find T. What is the procedure to solve it?

If I am required to compute the full transformation matrix compromising of the following sequence of operations:
rotation by 30 degrees about x-axis
translation by 1, -1, 4 in x, y and z, respectively
rotation by 45 degrees about y axis
Can I compute the rotation, translation and rotation matrix or would I be required to compute the rotation, rotation and translation matrix?

The problem is this:
The impulse response of a system is the output from this system when excited by an input signal $\mathrm{\delta <!--\; \delta \; -->}(k)$ that is zero everywhere, except at $k=0$, where it is equal to 1. Using this definition and the general form of the solution of a difference equation, write the output of a linear system described by:
$y(k)–3y(k–1)–4y(k–2)=\mathrm{\delta <!--\; \delta \; -->}(k)+2\mathrm{\delta <!--\; \delta \; -->}(k–1)$
The initial conditions are: $y(–2)=y(–1)=0$.
My question is: How can the particular solution be found using the method of undetermined coefficients if the non-homogeneous equation is also a difference equation?

I know that matrix multiplication is not commutative. And yet, I am taught that transformation matrices can be composed to arrive at a single transformation. So then, suppose that I have 3 matrices - each representing a rotation about each axis of a 3D coordinate system. How can I be sure that I arrive at the correct result, if the order matters?

1) Suppose $B=(w_1,w_2,\dots <!--\; \dots \; -->,w_k,v_1,\dots <!--\; \dots \; -->,v_{n-<!--\; -\; -->k})$ is a basis for $V$ and $W$ = Span $(w_1,\dots <!--\; \dots \; -->,w_k)$ is T-invariant. What does the matrix of $T$ with respect to $B$ look like?
2) Suppose $V$ is the direct sum of $W_1$ and $W_2$, with $W_1,W_2$ both T-invariant. If $B=(u_1,u_2,\dots <!--\; \dots \; -->,u_n,z_1,z_2,\dots <!--\; \dots \; -->,z_n)$ is a basis for $V$ with the $u$'s being a basis for $W_1$ and the $z$'s being a basis for $W_2$, what does the matrix of $T$ with respect to $B$ look like?

The problem reads:
The solution of a certain differential equation is of the form
$y(t)=a\exp <!--\; \; -->(5t)+b\exp <!--\; \; -->(8t)$
where $a$ and $b$ are constants.
The solution has initial conditions $y(0)=5$ and $y^{\prime }(0)=5$
Find the solution by using the initial conditions to get linear equations for $a$ and $b$. ....................
What I did was solve using the initial conditions and I found that
$a+b=5$
and $5a+8b=5.$
Am I totally on the wrong track? I don't know what it means to find a linear equation for $a$ and $b$. I'd appreciate it if you could solve it step by step.

$F:P_2\text{ to }{P}_2$ to $F:P_2\text{ to }{P}_2$ with the conditions that $F(p_0)=f$, $F(p_1)=g$, $F(p_2)=h$ where $f(x)=x^2+3$, $g(x)=x^2-<!--\; -\; -->x$, $h(x)=2+x$. Find the transformation matrix for $F$ in the basis $(p_0,p_1,p_2)$

Linear transformation and its matrix
two bases:
$A=\{v_1,v_2,v_3\}$ and $B=\{2v_1,v_2+v_3,-<!--\; -\; -->v_1+2v_2-<!--\; -\; -->v_3\}$
There is also a linear transformation: $T:{\mathbb{R}}^3\to <!--\; \to \; -->{\mathbb{R}}^3$
Matrix in base $A$:
$M_T^A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 1& 1& 0\end{array}\right]$
Now I am to find matrix of the linear transformation $T$ in base $B$.
I have found two transition matrixes (from base $A$ to $B$ and from $B$ to $A$):
$P_A^B=\left[\begin{array}{ccc}2& 0& -<!--\; -\; -->1\\ 0& 1& 2\\ 0& 1& -<!--\; -\; -->1\end{array}\right]$
$(P_A^B{)}^{-<!--\; -\; -->1}=P_B^A=\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{6}& \frac{-<!--\; -\; -->1}{6}\\ 0& \frac{1}{3}& \frac{2}{3}\\ 0& \frac{1}{3}& \frac{-<!--\; -\; -->1}{3}\end{array}\right]$
How can I find $M_T^B$?

struggling on the properties of the idempotent matrix, namely for any $n\times <!--\; \times \; -->n$ matrix, $A^2=A.$. The projection matrix defined by $M=I_n-<!--\; -\; -->A{\left(A^{T}A\right)}^{-<!--\; -\; -->1}{A}^T$ is an idempotent matrix. The question is, for any given $n\times <!--\; \times \; -->m$ ($n>m$) matrix $B,$, do we have
$\begin{array}{rcl}M& =& I_n-<!--\; -\; -->A{\left(A^{T}A\right)}^{-<!--\; -\; -->1}{A}^T\\ & =& I_n-<!--\; -\; -->AB{\left(B^T{A}^{T}AB\right)}^{-<!--\; -\; -->1}{B}^T{A}^T,\end{array}$
since $AB$ is basically the linear transformation of matrix $A.$

A linear transformation acts on $R^2$ by first doubling the x-coordinate and then rotating the plane by an angle of $\mathrm{\pi <!--\; \pi \; -->}/2$ (${90}^{\circ <!--\; \circ \; -->}$) counter-clockwise. What is the corresponding matrix to this linear transformation?

For a transformation
$F(X)=X^T$
on $V=M_{3,3}(\mathbb{R})$, apparently the matrix of $F$ is a $9\times <!--\; \times \; -->9$ matrix. How can this be possible? Isn't the definition that
$F(X)=AX?$
, so if $A$ is the $9\times <!--\; \times \; -->9$ matrix, we can't multiply a $9\times <!--\; \times \; -->9$ matrix with a $3\times <!--\; \times \; -->3$, can we?

Let $T_1$ be a reflection of ${\mathbb{R}}^3$ in the xy plane, $T_2$ is a reflection of ${\mathbb{R}}^3$ in the xz plane. What is the standard matrix of transformation $T_2{T}_1$?
Here's my thinking so far:
Since the standard matrix for reflections in xy is
$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$
Similarly, standard matrix for orthogonal projection in the xz plane is
$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]$
I could multiply
$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]\ast <!--\; \ast \; -->\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$
to yield
$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$
Could someone confirm for me if this is a valid approach?

Let $\{x,y,z\}$ be the basis of ${\mathbb{R}}^3$ and $A:{\mathbb{R}}^3\to <!--\; \backslash rightarrow\; -->{\mathbb{R}}^3$ is a linear operator given by the matrix:
$\left[\begin{array}{ccc}-<!--\; -\; -->1& -<!--\; -\; -->2& -<!--\; -\; -->5\\ 0& 2& 2\\ 2& 2& 4\end{array}\right]$
a) Determine the transformation matrix $A$ with respect to the basis $\{x+y,y+z,z+x\}.$
b)Determine if A is a linear transformation.
$A$
What is the process in determining the transformation matrix with respect to a new basis when you already have a transformation matrix with respect to another?

The homogenous representation of a circle is given by $x^2+y^2+2gxz+2fyz+c{z}^2=0$ (or, equivalently, if we set $z=1$, $x^2+y^2+2gx+2fy+c=0$). Now, given 3 points (in a homogenous form), we can solve a system of linear equations and retrieve the unknowns $f$, $g$ and $c$.
This is all very nice (because of linear algebra), but what do these unknowns actually represent with respect to the circle? Which of these numbers represent the x and y coordinates of a circle and which one represents the radius?
Apparently, $-<!--\; -\; -->f$ and $-<!--\; -\; -->g$ would be the $x$ and $y$ coordinate of the center of the circle? Why is that the case? I would like to see a proof/derivation of it. Also, what is the radius then?

Matrix transformation conserving the "positive semi-definite" aspect
Let's say I have two covariance matrices $A$ and $B$ (so they're both positive semi-definite), What kind of transformations can I apply on either one of them or both without loosing the positive-definite aspect in the resuling matrix ?